Last time we made progress on figuring out how high a tennis ball can bounce in the classic experiment where we drop the tennis ball on top a basketball. We didn’t find the answer, but we said that if the tennis ball picks up a speed in falling, then immediately after bouncing off the basketball, it could have a maximum upward speed of .

Today we want to figure out what that means in terms of how high the tennis ball will bounce. It turns out that the tennis ball does not bounce three times as high as it started when it rebounds with three times the speed. In fact it bounces much higher.

After bouncing off the basketball, the tennis ball rises, but slows down under the influence of gravity until it comes to a stop at the top of its trajectory. To understand how high it goes, we must answer the question, “what does the influence of gravity do to the motion of the ball?”

One of the first people to understand this question and its answer was Galileo (although several people came to the correct conclusion before him). We’ll look at a few passages of his famous book, Dialogue Concerning Two New Sciences. (specifically this part)

Galileo begins by stating that he thinks “uniformly accelerated motion”, the motion of a tennis ball thrown into the air, should be very simple.

When, therefore, I observe a stone initially at rest falling from an elevated position and continually acquiring new increments of speed, why should I not believe that such increases take place in a manner which is exceedingly simple and rather obvious to everybody? If now we examine the matter carefully we find no addition or increment more simple than that which repeats itself always in the same manner.

In other words, the way the speed of a falling body changes shouldn’t depend on how high it is, or how long it’s been falling, or how far it’s fallen. It should depend on nothing at all – be always the same.

This may be a lot to swallow, but let’s look at one good reason (not due to Galileo) that we might expect the way gravity acts on a falling object not to change with how high the object is above the Earth’s surface. The radius of the Earth is very large compared to the heights we throw things. We expect that if the effects of Earth’s gravity do change with your distance from the center of the Earth, they ought to do so on a distance scale roughly equal to the radius of the Earth.

That is, if you want a significant difference in the force of gravity, you ought to change your position by something significant compared to the radius of the Earth, since it defines the only natural length scale in this problem. The radius of the Earth is roughly six million meters, so throwing a tennis ball up in the air six meters is completely negligible. We could calculate the effects of gravity using Newton’s gravitational law, but that is unnecessary. Any other reasonable gravity law ought to work out basically the same. Near the surface of the Earth, your height should not affect how gravity acts on you.

This is only one part of what Galileo said. For example, he also believes that how fast an object moves should not affect how gravity acts on it. This belief may have been stimulated by the relativity principle – that all laws of physics should be the same, even when you’re moving. Relativity does not absolutely preclude a force that depends on velocity, though (magnetic forces do this), but velocity-dependent forces are not as simple as velocity-independent forces, and for the time being Galileo is guessing that the way gravity acts ought to be very simple.

We continue with the G-spot’s wise words:

A motion is said to be uniformly accelerated, when starting from rest, it acquires, during equal time-intervals, equal increments of speed.

This is Galileo’s working idea of how things fall. If you drop something, and at the end of one second it goes speed , then at the end of two seconds it will be going , and at the end of three seconds , etc. A plot of speed over time, if we drop an object from rest, should look like this:

Now that we have a theory for what the speed of the tennis ball does, we should be able to figure out how high it goes. The tennis ball reaches its highest height when its speed is zero, so we simply need to keep track of its speed until that speed falls to zero. If we know how fast it was going and for how long, we should also know how far it traveled.

I’ll paraphrase Galileo’s arguments here rather than quoting them, since he does not directly answer our exact question. The relevant pages are 171 – 178.

First, let us suppose it takes the tennis ball a time to fall before bouncing, and it acquires speed in that time. We know it bounces back up with speed . It loses speed in the same way it gained speed – the same amount per second. So after a time , the ball loses speed , and is down to moving at speed . The ball comes to a stop at the height of its trajectory after a time .

To summarize, if the ball gains and loses the same amount of speed in any moment of time, then if it two balls bounce upward, one three times as fast as the other, the fast one will take three times as long to get reach its apex.

The distance the ball travels just , which is the green area shaded in the previous drawing.

Here is a plot of the speed of the ball as it rises:

It rises three times as long as it fell, and the distance it rises is purple the area in the above chart. Laying the two plots together, we see that the purple area is nine times as large as the green one – three times taller and three times wider.

Now we have our first answer to how high a tennis ball can bounce when dropped on top a basketball. It can bounce nine times as high, when we make the following assumptions:

- When things bounce off the ground, they change their direction keeping exactly the same speed and hence bounce back to the same height. (first post).
- A basketball is so much bigger than a tennis ball that it essentially acts as the ground – the tennis ball bounces off just the same as it would bounce off the ground. (third post)
- To understand the way something in motion works, we can imagine we are moving alongside it at the same speed so that it isn’t moving from our point of view, and understand it that way. Then we can imagine going back to the frame in which the thing is moving and translating over our new knowledge over. (third post)
- Gravity pulls an object down such that it gives it the same additional amount of speed in each moment of time. (this post)

My original claim was that I could have understood all these ideas as a child. I think that’s right. I was a pretty bright kid, and if someone had sat down to explain this reasoning to me, and answered my questions, I think I’d have gotten it. But I also hope I’d have realized there’s a problem. When you actually do the experiment, the tennis ball doesn’t bounce nine times as high, or anywhere near that. Three times as high is pretty good for this experiment. So I’d like to think I’d have noticed that, and asked for an explanation of the discrepancy.

We began to discuss this in part two, where we looked at why things bounce to a lower height than they’re dropped from. The assumption about reversing direction and speed when bouncing is simply not correct. It is also not correct to assume that the basketball is so much larger than the tennis ball that it acts like the ground, but this is a smaller source of error. It isn’t true that gravity is completely uniform, either, or that the only influence on the falling ball is from gravity. We’ll look at these things in more detail in a later post.

Before doing that, though, the next post or two will continue looking at the passage from Galileo. This passage isn’t interesting to me simply because it is an early source of someone understanding this fairly simple problem. It’s interesting because it’s an illustration of Galileo laying down a more sophisticated understanding of how we can understand nature. I want to look at what Galileo did and didn’t know, but also at how much he understood about what he did and didn’t know, and how he came to his conclusions.

There’s also a very surprising and egregious logical error in the passage, so we’ll talk about that, too, before returning to the tennis ball a little down the line.